In particular, \(\mathrm {SL}(n,\mathbb {C})=\lbrace A\in \mathrm {GL}(n,\mathbb {C})\colon \det (A)=1\rbrace \) . Due to \(G=\lbrace A_\sigma \colon \sigma \in \mathrm {DGal}(L/K)\rbrace \subset \mathrm {GL}(n,\mathbb {C})\) , we see that \(\mathrm {DGal}(L/K)\)  has a faithful representation as an algebraic group of matrices in where \(G^0\)  denotes the connected identity component of \(G\)  (the biggest algebraic connected subgroup of \(G\) ). In this terminology, we say that a linear differential equation is integrable in the sense of differential Galois theory whether the connected identity component of its differential Galois is a solvable group. Moreover, this definition of integrability leads to the obtaining of solutions in closed form if and only if \(G^0\)  is solvable, see [1]} for full explanation and details. From now on, integrable in this paper means integrable in terms of differential Galois theory, see [2]}.
